From MusicWeb-International.com:

I'll give you £100 if you can show me the words and music of 'a Marie Lloyd song called She Sits Among The Cabbages and Peas or Leeks' It may be a good story but it just aint true.
Max Tyler.
Historian and Archivist. British Music Hall Society

The story he was reacting to was this:

Marie [Lloyd] once had a song about gardening called "I Sits Among My Cabbages and Peas" but when the Lord Chamberlain, effectively the theatre censor, saw the words he was outraged. Such vulgarity would not be allowed, she was told by his office. Quick as a flash, Marie changed the words and the song became "I Sits Among My Cabbages and Leaks". The Lord Chamberlain's blushes were spared, the crowd roared and Marie Lloyd marched on. Soon she reverted to the original, of course.

Different versions of the story have circulated, and can be found on the Internet, and at Mudcat. Sometimes it's "…Lettuces and Leaks."

Unfortunately, I can find no information about the origin of the song that Neil Hamburger sings. I suspect it's a recent composition inspired by the Marie Lloyd story.

However, I did find this, at immortalia.com:

A RURAL BALLAD

Tempt me not away to mountains, to lakes or shady dell,
Nor to the sea shore where the sands are full of fleas;
But take me [to] the country, the place I love so well,
'Mid the hay fields, and the cabbage and peas.

Tempt me not with lovely ladies, with their husbands left at home,
And a bunch of bathers basking at their knees;
I prefer the dairy maiden upon her native loam,
In the land where grows the cabbages and peas;

Where the hay upon the meadow gives a fragrance to the air;
Where the cow and calf are browsing 'neath the trees;
In the golden glow of evening I would greet the maiden fair,
As she sits among the cabbages and peas.

--from "The Book of a Thousand Laughs," by "O. U. Schweinickle," 1928.